Method and apparatus for generating surfaces of revolution

ABSTRACT

Method and apparatus for generating, on optical material or metal, aspheric surfaces of revolution. Basically, the subject invention includes a lever arm pivotally mounted at one end portion thereof for simultaneous movement in at least two mutually orthogonal planes. The end portion of the lever arm remote from the pivot is adapted to ride upon and follow a cam surface so as to impart a predetermined motion to the lever arm. A tool is mounted on the lever arm, a predetermined distance from the pivot, and is adapted to rotate about an axis extending through the pivot point of the lever arm. A work blank is suitably positioned beneath the lever arm for rotation about a vertical axis located a predetermined radial distance from the pivot point so that movement of the end portion of the lever arm in contact with the cam will be a linear amplification of the movement of the lever arm and, hence, the tool with respect to the work blank. The surface of the tool is adapted to make tangential contact with the surface of the work blank at all times and the blank may be vertically fed to control material removal.

Baker sate Eye-6 1,065

May 23, 1972 Inventor:

Assignee:

Filed:

Appl. No.:

James G. Baker, Winchester, Mass.

Polaroid Corporation, Cambridge, Mass.

May 1, 1970 ..B24b 17/04, B24b l/00 References Cited UNITED STATES PATENTS Volk Ford De Vlieg.. Lanius ..51/284 Primary Examiner-Donald G. Kelly AttorneyBrown and Mikulka, William D. Roberson and Michael Bard ABSTRACT Method and apparatus for generating, on optical material or metal, aspheric surfaces of revolution. Basically, the subject invention includes a lever arm pivotally mounted at one end portion thereof for simultaneous movement in at least two mutually orthogonal planes. The end portion of the lever arm remote from the pivot is adapted to ride upon and follow a cam surface so as to impart a predetermined motion to the lever arm. A tool is mounted on the lever arm, a predetermined distance from the pivot, and is adapted to rotate about an axis extending through the pivot point of the lever arm. A work blank is suitably positioned beneath the lever arm for rotation about a vertical axis located a predetermined radial distance from the pivot point so that movement of the end portion of the lever arm in contact with the cam will be a linear amplification of the movement of the lever arm and, hence, the tool with respect to the work blank. The surface of the tool is adapted to make tangential contact with the surface of the work blank at all times and the blank may be vertically fed to control material removal.

20 Claims, 5 Drawing Figures sf ICfIC- lens s urfac cs in-f fit 68 "M 111111 m lllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllll Patented May 23, 1972 3,664,065

4 Sheets-Sheet 1 lllllmm Mum INVENTOR. JAMES G. BAKER ATTORNEYS Patented May 23, 1972 3,664,065

4 Sheets-Sheet 13 INVENTOR. JAMES G. BAKER Ema/4a Mad m and mm 641/10! ATTORNEYS Patented May 23, 1972 4 Sheets-Sheet 4 INVENTOR. JAMES G. BAKER l/wwn and m and 7725M EM ATTORNEYS BACKGROUND OF THE INVENTION The instant invention relates generally to the art of forming surfaces of peculiar contours and more particularly to an improved method and apparatus to be used for the purpose of generating aspheric surfaces of revolution on optical material or metal.

In lens grinding, generally, and particularly in the generation of aspheric surfaces, very fine accuracy control of the grinding tool is necessary. Typically, prior art devices for generating aspheric surfaces have utilized profile cam arrangements for causing the tool to follow a prescribed path. Devices of the prior art have frequently utilized tools approximating a single point and rotating about an axis generally parallel to that of the work piece. Altemately, prior art aspheric surface generators utilized tools mounted for rotation about a generally horizontal axis which was made to follow a cammed surface while remaining parallel to its initial position.

In both of the above-described prior art systems, accuracy of control of the work surface suffered to the extent that the work invariably had to be hand finished and polished.

SUMMARY OF THE INVENTION The subject invention provides a cam following generator and procedure for providing optical material, metal, or the like with aspheric surfaces of revolution. Basically, the invention envisions a lever arm which is pivotally mounted at one end thereof so as to move freely in at least two mutually orthogonal planes. A work holder supports a work piece (comprising optical material, metal or the like) for rotation about a vertical axis and the work holder is positioned with respect to the pivot point of the lever arm so that the perpendicular distance from said pivot point to said vertical axis is a precise predetermined amount.

A frusto-conical shape tool is mounted on the lever arm for rotation about an axis extending through said pivot point, with its outer periphery converging towards said pivot point and with the lowermost portion of its surface parallel to the longitudinal axis of said lever arm. The work holder is adapted to move vertically toward and away from said tool which, in the case of a convex surface, will always contact the work tangentially.

The end portion of said lever arm remote from said pivot is provided with a cam follower and rides upon a profile cam which, in turn, is curved about a vertical axis passing through said pivot point such that the radius of curvature of the cam is the perpendicular distance from the cam to the vertical axis passing through said pivot point.

In operation, the work piece is brought into engagement with the tool and both are caused to rotate about their respective axes. An abrasive slurry is provided between the tool and the work piece so that the tool may wear away selected portions of the surface of the work piece. The lever arm is caused to pivot by causing the portion of said lever arm remote from said pivot point to follow the surface of said cam. By virtue of the geometry of the system, the tool will traverse a path which is a linear submultiple of the profile ofthe cam and, hence, the profile of the cross section of the work piece will conform to that of said cam on a reduced scale dependent on the ratio of the distance of the vertical axis of rotation of the work piece to the pivot to the radius of curvature of said cam.

In consonance with the foregoing, it is desired to effect an improved method and apparatus for generating aspheric surfaces of revolution.

Further, it is an object of the present invention to provide a simpler and less expensive means for generating aspheric surfaces of revolution than was heretofore available.

Another object of this invention is to provide an improved cam following generator and method to be used for the purpose of generating aspheric surfaces of revolution on optical material, metal, or the like.

Yet another object of the instant invention is to provide an improved method and apparatus for generating aspheric lenses or the like by imparting a predetermined movement to one 4 end portion of a pivoted lever arm and positioning a tool on said lever arm a predetermined distance from said pivot, whereby the movement of said end portion will be a predetermined arnplification of the movement of said tool.

It is a further object of the present invention to provide an improved cam following generator and method for forming aspheric surfaces of revolution on optical material without the need of further polishing.

Other objects, and many of the attendant advantages of the present invention, will be better appreciated and said invention will become clearly understood with reference to the following detailed description when considered in conjunction with the accompanying drawings illustrating one embodiment of the instant invention, wherein:

FIG. 1 provides a fragmentary perspective of the subject invention partly in section;

FIG. 2 provides a top plan of the invention of FIG. 1;

FIG. 3 provides a fragmentary perspective of a portion of the invention illustrated in FIG. 1 partly in section and particularly illustrating the feed mechanism for the work piece;

FIG. 4 provides a partial perspective of an alternate tool to be used in the apparatus of FIG. 1; and

FIG. 5 provides a graphical representation of the contact path followed by a frusto-conical tool rotating about an axis, which in turn is pivoting about a point, and rolling over the surface of a sphere.

Referring to the drawings in more detail and more particularly to FIG. 1, the subject invention is shown generally at 2 and includes a rectangular base 4 supported by legs 6. A pivot support 8 is secured to one end portion of the rectangular base 4 and is provided with a spherical recess 10 in the upper end portion thereof and centered on its longitudinal axis. A cap 12 having a generally cylindrical opening 14 extending longitudinally therethrough is threadably secured to said pivot support 8 over said spherical recess 10.

A lever arm 16, having a longitudinal axis 18, is pivotally secured to said pivot support 8 by means of a pivot pin 20. The pivot pin 20 is of generally cylindrical shape and extends through one end portion 22 of the lever arm 16, normal to the longitudinal axis 18, to a ball shape end portion 24 which is received within the spherical recess 10. The cylindrical opening 14 is of sufficiently greater diameter than the pivot pin 20 to permit the lever arm 16 to pivot on the ball shape end portion 24. Also, the pivot pin 20 is fixedly secured to the lever arm 16, as by means of a set screw (not shown).

A profile cam 26 is secured to the rectangular base 4 remote from said pivot support 8 and, as best seen in FIG. 2, is curved, in the plane of said base 4, about an axis passing through the center of the ball shape end portion 24 of the pivot pin 20 and perpendicular to said base 4.

A cam follower 28 is rotatably secured to the end portion of said lever arm 16 remote from said pivot support 8 and is perpendicular to a line extending from the center of the ball shape end portion 24 to the center of rotation of said cam follower 28. A handle 30 is secured to said lever arm 16, with its longitudinal axis coaxial with the axis of rotation of the cam follower 28, and serves to allow the user to cause said cam follower 28 to roll along the periphery 32 of the profile cam 26 thus causing one end portion of the lever arm 16 to follow the curvature of said periphery 32.

A support fixture 34 is secured normal to said base 4 adjacent said pivot support 24 and a bearing pin 36 passes transversely through the upper portion thereof and is secured therein generally parallel to said base 4.

The upper portion of the pivot pin 20 is secured between said bearing pin 36 and a second bearing pin 38 by means of a clamp 40 such that said bearing pin 20 may rotate about its said longitudinal axis and pivot in a plane passing through its longitudinal axis and parallel to the longitudinal axis of the bearing pin 36. The clamp 40 is provided with a pair of spaced parallel J-shape legs 42 which engage opposite end portions of the bearing pin 36, permitting the clamp 40 to rock thereon. The end portion of the clamp 40 remote from said legs 42 is provided with a U-shape recess 44 adapted to receive a clamping screw 46 which passes therethrough and is threaded into the support fixture 34. The bearing pin 38 is loosely supported on the upper surface of the legs 42 such that tightening of the clamping screw 46 will cause the clamp 40 to rock about the bearing pin 36 clamping one end portion of the pivot pin 20 between the bearing pins 38 and 36.

A support shaft 48 extends axially outward from the pivoted end portion 22 of the lever arm 16 and supports a counterweight 50 which is slideable therealong and serves to counterbalance the lever arm 16 and its associated structure. A set screw 52 is provided for securing the counterweight 50 at a desired location along the shaft 48 so as to effectively regulate the force that must be applied to the handle 30 to effect movement of the lever arm 16.

As best seen in FIGS. 1 and 3, a spindle 52 is rotatably secured within a housing 54 normal to the rectangular base 4 and extends outwardly of said housing 54, below said base 4, to a pulley 56 at its lower end and extends to a work support 58 at its upper end. The spindle 52 is so positioned that its longitudinal axis is a predetermined distance from the center of curvature of the ball shape end portion 24 of the pivot 20 and, hence. from the pivotal center of the arm 16.

A work blank 60 which may comprise a generally spherical lens blank, or other optical or non-optical material, is positioned on said work support 58 with its center of curvature along the longitudinal axis of the spindle 52. A motor 62 havr ing a shaft 64 and a pulley 66 serves to rotate the spindle 52 and the work blank 60 via a belt 68 which passes over the pulley 56 and the pulley 66.

As best seen in FIG. 3, the housing 54 is secured to a vertical slide 70 which is dovetailed to a vertical support 72 and adapted to slide vertically therealong, together with the spindle and work support 58. The movement of the vertical slide 70 along the support 72 may be precisely controlled as by means of a rack (not shown) secured to the vertical slide 70 and a worm (not shown) secured to the support 72 and driven by a shaft 74 which is coupled through a pair of bevel gears 76 and 78 to a shaft 80 and crank assembly 82 (FIG. 1). A dial indicator 84 is secured to the housing 54 via a dog 86 and serves to indicate the vertical position of the housing 54 and, hence, the spindle 52 and the work blank 60, with respect to the rectangular base 4.

A shaping tool 88 is secured to a shaft 90 for rotation therewith and said shaft 90 is rotatably secured to said lever arm 16 in a well-known manner such that said shaping tool 88 will be positioned a predetermined distance from the pivot pin 20. The shaping tool 88 is rotatably driven with its shaft 90 by a motor 92, via a motor shaft 94 and a pair of couplings 96 and 98, about an axis extending through the center of the ball shape end portion 24 of the pivot pin 20 and hence the pivot point ofthe lever arm 16. I

The periphery 32 of the cam 26 is so formed that, with the lever arm 16 in the position shown in FIGS. 1 and 2, the shaping tool 88 will be centered over the work blank 60 which, in turn, may be vertically fed into engagement with the shaping tool 88 by driving the vertical slide 70 with the crank 82.

As best seen in FIG. 1, for the case ofa convex work blank 60, it has been found beneficial to utilize a frusto-conical shaping tool 88 so positioned on the lever arm 16 that its lowermost edge is parallel to the longitudinal axis 18 of said lever arm 16. Shaping tools of other configurations such as a cylindrical shape too] would be efficacious but would necessitate that a cam 26 having a periphery 32 ofa different profile be used.

Before proceeding further, reference may be had to FIG. for a simplified representation of some of the geometrical principles inherent in the subject invention. Referring to FIG. 5 in more detail, a hemisphere 98 is shown mounted in a support 100 in spaced relation to a pin 102. A frustum 104 of a cone is shown rotatably mounted on one end portion of a shaft 106 which in turn is pivotally connected to the pin 102. The frustum 104 is of such shape and is so positioned that it initially contacts the apex of the hemisphere 98 such that the edge of such frustum 104 in contact with the hemisphere 98 is tangential thereto.

If we now pivot the shaft 106 about the pin 102 while rolling the frustum 104 along the surface of the hemisphere 98, it is seen that the path 108 so traced will be a circle lying on the surface of the hemisphere 98 and normal to all lines radiating thereto from the pivot point ofthe frustum 104.

It follows, therefore, that the point of contact between the frustum 104 and the hemisphere 98 will trace a circular path about the periphery of said frustum 104 of a diameter equal to the diameter of said frustum 104 at its point of contact with the apex of said hemisphere 98. Thus, the frustum 104 will be in contact with the hemisphere 98 at some point along the surface thereof and always a fixed distance from the pivot point ofsaid frustum 104 at all times.

It should be noted at this point that where the frustum 104 is replaced with a cylinder having the same axis of rotation and pivoted at the same point, the path traced by the point ofcontact between the cylinder and the hemisphere 98 will also be a circle, albeit a different one from that traced by the frustum 104.

Referring again to FIG. 1 and considering the case of a hemispheric convex work blank 60, the cam follower 28 is initially positioned on the periphery 32 of the cam 26 such that the shaping tool 88 is centrally positioned over the work blank 60. Clockwise rotation of the crank 82 will bring the work blank 60 into engagement with the lowermost edge of the shaping tool 88 which, looking in the direction of its increas ing diameter, is caused to rotate clockwise by the motor 92. Typically, the surface of the shaping tool 88 is smooth and an abrasive slurry may be manually applied between said tool 88 and the work blank 60 in a well-known manner to effect the abrasive removal ofsome of the material of the work blank 60.

Looking down on said work blank 60, it will be seen to be rotating clockwise by virtue of the driving torque supplied by the motor 62. It is apparent that the direction of rotation of the shaping tool 88 and the work blank 60 are such that the abrasive slurry will not be thrown back at the operator. When sufficient material has been removed from the surface of the work blank 60, at the initial point of contact, the lever arm 16 is pivoted on the pivot pin 20 by moving the cam follower 28 along the periphery 32 of the cam 26 until the work blank 60 is made to conform to the shape dictated by the profile of said cam 26.

It will be recalled in connection with our discussion of the geometric representation of FIG. 5, that the distance from the pivot point of the lever arm 16 to the point of contact between the tool 88 and the work blank 60 is constant. Obviously, the distance from the pivot point of the lever arm 16 to the center of rotation of the cam follower 28 is also constant. Ac cordingly, the finished work blank 60 will conform to the shape of the periphery 32 of the cam 26 on a predetermined reduced scale dependent on the ratio of the distance between the pivot point of the lever arm 16 and the point of contact between the shaping tool 88 and the work blank 60 to the distance between said pivot point and the center of rotation of the cam follower 28. Because the periphery 32 of the cam 26 is a very much magnified representation of the shape it is desired to effect in the work blank 60, extremely precise con trol of the shaping of the surface of the work blank 60 may be effected with the instant invention.

At this point, it should be noted that, although the operation of the instant invention has been described with reference to a convex hemispheric work blank 60 and a frusto-conical shaping tool 88, the teachings herein are equally applicable to a variety of tool shapes such as a cylindrical tool, as well as to work blank surfaces which are, for example, concave.

In the case where the work blank is generally concave in shape, it has been found expedient to utilize a shaping tool 110 which, as best seen in FIG. 4, is provided with a periphery 1 12 which comprises a section of a toroid. When generating concave surfaces of revolution, it has been found desirable to provide the periphery 112 of the tool 110 with a radius of curvature about an axis 114, and within a plane 116 perpendicular to the axis 114 and passing through the center of curvature 118 of said periphery 112, which is smaller than the smallest radius of curvature to be generated. Likewise, said periphery 112 is provided with a radius of curvature about said center of curvature 118, within a plane 120 containing said axis 114 and normal to the plane 116, which is also smaller than the smallest radius of curvature to be generated.

It can readily be seen that many variations and modifica tions of the present invention are possible in the light of the aforementioned teachings, and it will be apparent to those skilled in the art that various changes in form and arrangement of components may be made to suit requirements without departing from the spirit and scope of the invention. It is therefore to be understood that within the scope of the appended claims, the instant invention may be practiced in a manner otherwise than is specifically described herein.

WHAT IS CLAIMED IS:

1. Apparatus for generating surfaces of revolution including:

a base;

a lever arm having a longitudinal axis and pivotally mounted, at one end portion thereof, on said base so as to be movable from an initial position to a position whereat said longitudinal axis will be oblique to its initial position;

means for securing a profile cam to said base remote from said pivotally mounted end portion;

means adapted to engage such cam and cause said lever to follow the contour of such profile cam;

a support secured to said base, vertically movable with respect thereto and rotatable about an axis nonnal thereto;

means for securing a work blank to said support;

a tool mounted on said lever arm a predetermined distance from the center of pivot thereof and rotatable about an axis extending through said center of pivot; and

means for bringing said work blank, into engagement with said tool to thereby remove selected portions of the surface of said work blank.

2. The invention as set forth in claim 1, wherein said tool is of such shape that a line drawn along the surface of said tool and through the intersection of said tool and said work blank will be tangent to said work blank.

3. A method for generating an aspheric surface of revolution on a work blank including the steps of:

rotating said work blank about an axis;

pivotally mounting a tool for traversing the surface of said work blank;

rotating said tool about an axis passing through the pivotal center thereof;

moving said work blank into engagement with said tool; and

traversing the surface of said rotating work blank with said rotating tool.

4. The method of claim 3, including the further step of causing said tool to follow a predetermined path in traversing said work blank differing from the shape ofsuch work blank.

5. The method as related in claim 4, wherein said predetermined path conforms to the cross-section of a desired aspheric surface of revolution.

6. Apparatus for generating aspheric surfaces including:

means for rotating a work blank about an axis;

a tool for removing selected portions from the surface of said work blank;

means for pivoting said tool about a point;

means for rotating said tool about an axis passing through said pivot point and said tool; means for axially moving said work blank into engagement with said tool; and means for causing said tool to traverse said work blank so as to remove a preselected amount of material from predetermined portions of the surface thereof.

7. The invention as recited in claim 6, wherein said tool pivoting means includes a lever arm pivoted about said point and further including means for causing the portion of said lever arm remote from said point to follow a predetermined path, whereby said tool will follow said path reduced in magnitude by a function dependent upon the ratio of the distance between the axis of rotation of said work blank and said pivot point to the distance between said pivot point and said lever arm portion remote from said pivot point.

8. The invention as related in claim 7, wherein said lever is pivoted at said point for movement about each of two mutually orthogonal axes.

9. The invention of claim 8, wherein the axis of rotation of said tool is oblique with respect to one of said orthogonal axes and is disposed at a variable angle with respect to the other of said orthogonal axes.

10. The invention related in claim 9, wherein said means for causing said lever arm to follow a predetermined path includes a cam, and further including a follower connected to said lever arm and adapted to engage said cam and traverse the contour thereof.

11. The invention as stated in claim 10, wherein said follower is connected to one end of said lever arm and is rotatable about an axis passing through said pivot point.

12. Means for generating an aspheric surface of revolution on a spherical work blank, including:

an arm having a longitudinal axis and pivotally supported at one end portion thereof so as to be movable from an initial position to a position whereat said longitudinal axis will be oblique to its initial position;

a tool rotatably secured to said arm;

means for effecting engagement between said work blank and said tool at a discrete point on said tool; and

means for causing said tool to traverse said work blank such that said discrete point will shift along the surface of said work blank and said tool.

13. The invention as described in claim 12, wherein said traverse causing means effects the shifting of said point along the surface of said work blank such that the locus of said points of engagement is a circle lying in a plane passing through the center of curvature of the work blank.

14. The invention as related in claim 12, wherein said point of engagement is a predetermined distance from the axis of rotation of said tool.

15. The invention as delineated in claim 14, wherein the periphery of said tool comprises a surface of revolution.

16. The invention as stated in claim 12, wherein said tool is frustum shape with its periphery extending to an imaginary apex at the center of pivot of said arm.

17. The invention as related in claim 2, wherein said tool is provided with a toroidal periphery.

18. The apparatus of claim 12, further including a cam and means for causing one end portion of said arm to follow the contour of said cam such that said tool will follow a path that is a predetermined function of the contour of said cam.

19. The invention as set forth in claim 18, wherein the contour of said aspheric surface of revolution is a predetermined function of that of said cam.

20. The invention as described in claim 12, additionally including means for effecting removal of a predetermined amount of material from the surface of said work blank by said tool. 

1. Apparatus for generating surfaces of revolution including: a base; a lever arm having a longitudinal axis and pivotally mounted, at one end portion thereof, on said base so as to be movable from an initial position to a position whereat said longitudinal axis will be oblique to its initial position; means for securinG a profile cam to said base remote from said pivotally mounted end portion; means adapted to engage such cam and cause said lever to follow the contour of such profile cam; a support secured to said base, vertically movable with respect thereto and rotatable about an axis normal thereto; means for securing a work blank to said support; a tool mounted on said lever arm a predetermined distance from the center of pivot thereof and rotatable about an axis extending through said center of pivot; and means for bringing said work blank, into engagement with said tool to thereby remove selected portions of the surface of said work blank.
 2. The invention as set forth in claim 1, wherein said tool is of such shape that a line drawn along the surface of said tool and through the intersection of said tool and said work blank will be tangent to said work blank.
 3. A method for generating an aspheric surface of revolution on a work blank including the steps of: rotating said work blank about an axis; pivotally mounting a tool for traversing the surface of said work blank; rotating said tool about an axis passing through the pivotal center thereof; moving said work blank into engagement with said tool; and traversing the surface of said rotating work blank with said rotating tool.
 4. The method of claim 3, including the further step of causing said tool to follow a predetermined path in traversing said work blank differing from the shape of such work blank.
 5. The method as related in claim 4, wherein said predetermined path conforms to the cross-section of a desired aspheric surface of revolution.
 6. Apparatus for generating aspheric surfaces including: means for rotating a work blank about an axis; a tool for removing selected portions from the surface of said work blank; means for pivoting said tool about a point; means for rotating said tool about an axis passing through said pivot point and said tool; means for axially moving said work blank into engagement with said tool; and means for causing said tool to traverse said work blank so as to remove a preselected amount of material from predetermined portions of the surface thereof.
 7. The invention as recited in claim 6, wherein said tool pivoting means includes a lever arm pivoted about said point and further including means for causing the portion of said lever arm remote from said point to follow a predetermined path, whereby said tool will follow said path reduced in magnitude by a function dependent upon the ratio of the distance between the axis of rotation of said work blank and said pivot point to the distance between said pivot point and said lever arm portion remote from said pivot point.
 8. The invention as related in claim 7, wherein said lever is pivoted at said point for movement about each of two mutually orthogonal axes.
 9. The invention of claim 8, wherein the axis of rotation of said tool is oblique with respect to one of said orthogonal axes and is disposed at a variable angle with respect to the other of said orthogonal axes.
 10. The invention related in claim 9, wherein said means for causing said lever arm to follow a predetermined path includes a cam, and further including a follower connected to said lever arm and adapted to engage said cam and traverse the contour thereof.
 11. The invention as stated in claim 10, wherein said follower is connected to one end of said lever arm and is rotatable about an axis passing through said pivot point.
 12. Means for generating an aspheric surface of revolution on a spherical work blank, including: an arm having a longitudinal axis and pivotally supported at one end portion thereof so as to be movable from an initial position to a position whereat said longitudinal axis will be oblique to its initial position; a tool rotatably secured to said arm; means for effecting engagement between said work blank and said tool at a discrete point on said tooL; and means for causing said tool to traverse said work blank such that said discrete point will shift along the surface of said work blank and said tool.
 13. The invention as described in claim 12, wherein said traverse causing means effects the shifting of said point along the surface of said work blank such that the locus of said points of engagement is a circle lying in a plane passing through the center of curvature of the work blank.
 14. The invention as related in claim 12, wherein said point of engagement is a predetermined distance from the axis of rotation of said tool.
 15. The invention as delineated in claim 14, wherein the periphery of said tool comprises a surface of revolution.
 16. The invention as stated in claim 12, wherein said tool is frustum shape with its periphery extending to an imaginary apex at the center of pivot of said arm.
 17. The invention as related in claim 2, wherein said tool is provided with a toroidal periphery.
 18. The apparatus of claim 12, further including a cam and means for causing one end portion of said arm to follow the contour of said cam such that said tool will follow a path that is a predetermined function of the contour of said cam.
 19. The invention as set forth in claim 18, wherein the contour of said aspheric surface of revolution is a predetermined function of that of said cam.
 20. The invention as described in claim 12, additionally including means for effecting removal of a predetermined amount of material from the surface of said work blank by said tool. 